A dartboard of radius 10 units and the wall it is hanging on are represented using a two-dimensional coordinat

Exercises · Q9

Q. A dartboard of radius 10 units and the wall it is hanging on are represented using a two-dimensional coordinate system, with the board’s center at coordinate (0,0). Variables x and y store the x-coordinate and the y-coordinate of a dart that hits the dartboard. Write a Python expression using variables x and y that evaluates to True if the dart hits (is within) the dartboard, and then evaluate the expression for these dart coordinates:

a) (0,0)

b) (10,10)

c) (6, 6)

d) (7,8)

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Concept understanding — Boolean Expressions

Boolean Expressions: From Intuition to Precision

Imagine a light switch. It has exactly two states: on or off. There is no dimmer, no half‑on, no third option. A Boolean expression is just a way of describing conditions that also have only two possible outcomes: true or false.

That’s the entire intuition. Every Boolean expression you will ever write is a question that the computer (or a circuit) answers with a clear yes or no.

The Building Blocks

A Boolean expression is built from three things:

Boolean variables – these are placeholders that can hold either true (1) or false (0). For example, A, B, rain, isWeekend.
Logical operators – these combine variables into questions. There are three fundamental ones:
- AND (written as $\cdot$ or $\land$ ): “Are both conditions true?”
- OR (written as $+$ or $\lor$ ): “Is at least one condition true?”
- NOT (written as $\overline{A}$ or $\lnot A$ ): “Is the condition false?”
Constants – true (1) and false (0).

Note

In Indian exam contexts (CBSE, ISC, JEE), Boolean algebra uses $+$ for OR, $\cdot$ for AND, and a bar over the variable for NOT. So $\overline{A} + B \cdot C$ means “(NOT A) OR (B AND C)”.

How They Work: The Truth Table

The simplest way to see what a Boolean expression means is to list every possible combination of its variables and compute the result. That list is called a truth table.

Take the expression $F = A + B$ (A OR B). There are $2^2 = 4$ possibilities:

$A$	$B$	$A + B$
0	0	0
0	1	1
1	0	1
1	1	1

The result is 1 whenever at least one input is 1. That’s the OR operation.

Now take $G = A \cdot B$ (A AND B):

$A$	$B$	$A \cdot B$
0	0	0
0	1	0
1	0	0
1	1	1

Only when both are 1 does the result become 1.

And NOT simply flips the value: $\overline{0} = 1$ , $\overline{1} = 0$ .

Tip

Think of AND as “strict” (both must be true) and OR as “generous” (any one true is enough). NOT is the “rebel” – it always says the opposite.

Why This Matters …

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A dart at (x, y) is on the board when its distance from the centre is at most the radius 10 — i.e. x² + y² ≤ 10², avoiding any square root:

x**2 + y**2 <= 100

Dart	x² + y²	Expression
(0, 0)	0	True
(10, 10)	200	False

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The dart is inside the board when its distance from the centre (0,0) is at most the radius 10, i.e. x**2 + y**2 <= 100 — True for (0,0) and (6,6), False for (10,10) and (7,8).

The concept is turning a geometric condition into a boolean (relational) expression. A point (x, y) lies within a circle of radius 10 centred at the origin when its distance from the centre is at most 10. Squaring both sides of distance <= 10 avoids the square root entirely:

x**2 + y**2 <= 100

This expression evaluates to a bool — exactly what the question asks for. Checking it for each dart:

for (x, y) in [(0, 0), (10, 10), (6, 6), (7, 8)]:
    print((x, y), x**2 + y**2 <= 100)

Part	(x, y)	x² + y²	Compared with 100	Value
a	(0, 0)	0 + 0 = 0	0 <= 100	True (dead centre — a bullseye)
b	(10, 10)	100 + 100 = 200	200 <= 100	False
c	(6, 6)	36 + 36 = 72	72 <= 100	True
d	(7, 8)	49 + 64 = 113	113 <= 100	False

Expected output:

(0, 0) True
(10, 10) False
(6, 6) True
(7, 8) False
``` …

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